Civil Engineering Reference
In-Depth Information
Since
δp
is arbitrary, one gets
∂p
r
∂n
[
Z
]
−
1
[
C
]
=−
{
p
r
}
(13.67)
p
a
Substituting Eq. (13.67) into Eq. (13.61) and recalling that on the interface
p
=
=
p
b
+
p
r
, the discrete form of Eq. (13.60) reads finally
1
jω
I
p
δu
s
n
[
C
]
{
[
C
]
T
u
s
n
}−
=
p
}+
δp
{
δp
[
A
]
{
p
−
p
b
}
(13.68)
where
1
jωρ
0
[
C
][
Z
]
−
1
[
C
]
[
A
]
=
(13.69)
is an admittance matrix. In consequence, the radiation of the porous medium into the semi
infinite fluid amounts to an admittance term added to the interface interstitial pressure
degrees of freedom and to additional interface coupling terms between the solid phase
and the interstitial pressure (first two terms in Equation 13.68). Note that the last term
involving
p
b
is the excitation term and disappears in the case of free radiation. In the
numerical implementation, the discretized form Equation (13.44) with
{
F
s
}=
0 combined
with Equation (13.68) leads to the following linear system
−
ω
2
[
M
]
+
[
K
]
−
[
C
]
+
[
C
]
u
s
p
0
F
f
[
H
]
ω
2
=
(13.70)
[
A
]
jω
−
[
C
]
T
[
Q
]
[
C
]
T
−
+
−
with
1
jω
[
A
]
F
f
{
}=
{
p
b
}
(13.71)
The system of equations (13.70) is first solved in terms of the porous solid phase
nodal displacements and interstitial nodal pressures. Next, the vibroacoustic indicators of
interest can be calculated.
13.9
Examples
13.9.1 Normal incidence absorption and transmission loss
of a foam: finite size effects
The first example considers an absorption problem and consists of a 0
.
5m
×
5
.
08 cm rectangular sample of foam backed by a rigid wall and excited by a plane
wave. Two load cases are considered: normal incidence (0
◦
,
0
◦
)
and oblique plane wave
(45
◦
,
0
◦
)
. The properties of the foam are given in Table 13.1. A mesh of 22
×
0
.
5m
15
linear brick poroelastic elements is used for the foam for the normal incidence while a
larger mesh of 32
×
22
×
15 elements is used for the oblique plane wave. These meshes
have been selected to ensure convergence of calculations. The meshing criteria are clas-
sical for the plate and are functions of the Biot's wavelengths for the foam (Chapter 6).
×
32
×
